Question

A manufacturer designs a cylindrical can with no top. The surface area of the can is 325cm². The can has base radius r centimetres and height h centimetres. Write a polynomial function to model the capacity, C cubic centimetres, of the can as a function of r.

Answer 1

The capacity C of a cylindrical can with a given surface area can be modeled as a function of radius r by calculating the volume V = πr²h, where h (height of the cylinder) is deduced from the relationship 2πrh + πr² = 325cm².

Step-by-step explanation:

To model the capacity C of a cylindrical can with no top and a surface area of 325cm² as a function of the base radius r centimeters, we can use the formula for the volume of a cylinder V = πr² h, where h is the height. Given that the surface area (which comprises the lateral area and the base area since there is no top) is 325cm², we can first find a relationship between h and r. The lateral surface area is given by 2πrh, and the base area (there is only one base since the can is open on top) is given by πr². Therefore, the sum of the base and the lateral surface area is equal to the given surface area:

πr² + 2πr h = 325cm²

Solving this equation for h gives us h as a function of r, which we can then substitute back into the volume equation to have the volume C solely in terms of r.